### Vector Calculus & Fourier Transforms

Vector Calculus & Fourier Transforms

UNIT-1

Vector calculus:  Vector Differentiation: Gradient — Directional derivative — Divergence — Curl — Scalar Potential. Vector Integration: Line integral — Work done — Area — Surface and volume integrals — Vector integral theorems: Greens, Stokes and Gauss Divergence theorems (without proof).

UNIT II:

Laplace Transforms: Laplace transforms of standard functions — Shifting theorems — Transforms of derivatives and integrals — Unit step function — Dirac’s delta function — Inverse Laplace transforms — Convolution theorem (without proof). Applications: Solving ordinary differential equations (initial value problems) using Laplace transforms.

UNIT III:

Fourier series and Fourier Transforms:  Fourier Series: Introduction — Periodic functions — Fourier series of periodic function — Dirichlet’s conditions — Even and odd functions — Change of interval — Half-range sine and cosine series.

Fourier Transforms: Fourier integral theorem (without proof) — Fourier sine and cosine integrals — Sine and cosine transforms — Properties — inverse transforms — Finite Fourier transforms.

UNIT IV:

PDE of first order: Formation of partial differential equations by elimination of arbitrary constants and arbitrary functions — Solutions of first order linear (Lagrange) equation and nonlinear (standard types) equations.